Delay-Probability-Distribution-Dependent ℋ_∞ FIR Filtering Design with Envelope Constraints

Abstract

This paper studies the problem of ℋ_∞ finite-impulse response (FIR) filtering design of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the requirement of pulse-shape is also considered in filter design. Employing the information about the size and probability distribution of delay, a delay-probability-distribution-dependent criterion is proposed for the filtering error system. Based on a Lyapunov-Krasovskii functional, a set of linear matrix inequalities (LMIs) are formulated to solve the problem. At last, a numerical example is used to demonstrate the effectiveness of the filter design approach proposed in the paper.

1. Introduction

In the studies about filtering problem, one most significant approach frequently applied in the past decades is Kalman filtering, the main idea of which is to minimize the variance of the estimation error assuming considered system dynamics to be exactly known and the external disturbances to be stationary Gaussian noises with known statistical properties [1, 2]. However, in many practical engineering applications, the statistical details about external noise are not available [3–8]. In these cases, many approaches are introduced to improve systems’ robustness, such as ℋ_∞, ℋ₂, and mixed ℋ_∞/ℋ₂ filtering [2, 9–13]. In this paper, the ℋ_∞ filtering approach is utilized.

On the other hand, time-delays are frequently encountered in practical engineering systems, such as manufacturing systems, power systems, and networked control systems [14–17]. Existence of delay makes the analysis and synthesis of systems a much more difficult task; meanwhile it is also the source of instability and poor performance in many cases [13, 18–20]. The main approaches to solve delay problems can be classified into delay-dependent approach and delay-independent approach. It has been shown in [21, 22] that the results obtained using delay-dependent approaches are generally less conservative than the delay-independent approaches ones [23]. Acknowledging this fact, the delay-dependent approach is applied in this paper.

In fact, the variation of delay may often stick to some probability distribution in spite of its varying and underivable property [24, 25]. Furthermore, in many real systems such as networked control systems, the time-varying delay may have some abrupt burst, leading to very large delay with a very small probability [26]. In this sense, the discussion about time-delay should not only depend on its size but also on its probability distribution. In this paper, a new filter design approach and new stability criteria for the filtering error system taking the stochastic characteristic of time-varying delay into account is proposed.

While an ℋ_∞ optimal filter can catch the frequency-domain property, the time-domain constraints such as envelope constraints or bounds on signals cannot be handled by this frequency-domain approach [27]. Among various time-domain specifications, envelope constraints, which make requirement on the pulse-shape, have significant applications in many practical engineering systems, such as communication systems, radar, sonar systems, and signal processing systems [28–31]. For instance, in deconvolution filtering and data channel equalization problems, it is extremely important to achieve a desired pulse-shape through designing an appropriate filter [27].

Therefore, aiming at incorporating both frequency-domain and time-domain constraints into the problem, we intend to design a filter satisfying the ℋ_∞ performance and subject to envelope constraints in outputs. Meanwhile, time-varying delays with certain stochastic characteristics in the transmission channel are also taken into account. With the proposed filter design approach, a more general condition of time-varying delay problem can be solved. As in most situations, although detailed and exact information about delay cannot be achieved, the delay’s probability distribution characteristics can be predicted or observed relatively easily. Once the probability information is gotten, the filter design approach can be developed.

In this paper, based on a Lyapunov-Krasovskii functional, we first present an ℋ_∞ optimal solution to the design of a finite-impulse response (FIR) filter using information about the range of time-varying delay and its probability distribution. Then, the envelope constraints are taken into consideration. The resultant filter is called an ℋ_∞ optimal Envelope-Constrained FIR (ECFIR) filter. We obtain the solution via solving an LMI optimization problem. At last, a numerical example is presented to illustrate the effectiveness of the proposed filtering design approach.

2. Problem Formulation and Preliminaries

Consider a filtering system shown in Figure 1, where Σ_𝑙 represents a linear dynamic system with state-space realization given by Σ_𝑙∶𝑥_𝑙(𝑘+1)=𝐴_𝑙𝑥_𝑙(𝑘)+𝐵_𝑙𝑤(𝑘)𝑠(𝑘)=𝐶_𝑙𝑥_𝑙(𝑘), (1) where 𝑥_𝑙(𝑘)∈ℜ^𝑛𝑙 is the model state vector, 𝑤(𝑘)∈ℜ^𝑛𝑤 is the input signal, 𝑠(𝑘)∈ℜ^𝑛𝑠 is the source signal generated by the model, and 𝐴_𝑙, 𝐵_𝑙, 𝐶_𝑙 are known constant matrices with appropriate dimensions. Then the output 𝑠(𝑘) is transmitted through a channel with time-varying delay modeled by Σ_𝑐∶𝑥_𝑐(𝑘+1)=𝐴_𝑐𝑥_𝑐(𝑘)+𝐴_𝑑𝑠(𝑘−𝑑(𝑘))+𝐵_𝑐𝑣(𝑘)𝑦(𝑘)=𝐶_𝑐𝑥_𝑐(𝑘)+𝐶_𝑑𝑠(𝑘−𝑑(𝑘))+𝐷_𝑐𝑣(𝑘), (2) where 𝑥_𝑐(𝑘)∈ℜ^𝑛𝑐 is the channel state vector, 𝑑(𝑘)∈[0,𝑑₂] is the time-varying delay with an upper bound of 𝑑₂, 𝑦(𝑘) is the output of the channel, and 𝑣(𝑘) is the disturbance input; 𝐴_𝑐, 𝐴_𝑑, 𝐵_𝑐, 𝐶_𝑐, 𝐶_𝑑, 𝐷_𝑐 are all known constant system matrices with appropriate dimensions. As is shown in (2), the source signal 𝑠(𝑘) suffers from influence of time-varying delay 𝑑(𝑘) and disturbance from the environment represented by 𝑣(𝑘). The output of transmission channel is 𝑦(𝑘), which is also the input signal of the filter. We are going to use the corrupted signal 𝑦(𝑘) to reconstruct original source signal.

Figure 1 

Filtering system.

Assumption 1. 𝑑(𝑘) changes randomly and for a constant 𝑑₁∈[0,𝑑₂], and the probability of 𝑑(𝑘)∈[0,𝑑₁) and 𝑑(𝑘)∈[𝑑₁,𝑑₂] can be known. The following sets and functions are defined: Ω₁=𝑘∶𝑑(𝑘)∈0,𝑑₁,Ω₂=𝑘∶𝑑(𝑘)∈𝑑₁,𝑑₂,𝑑₁(𝑘)=𝑑(𝑘),for𝑘∈Ω₁0for𝑘∉Ω₁,𝑑₂(𝑘)=𝑑(𝑘),for𝑘∈Ω₂𝑑₁,for𝑘∉Ω₂. (3) Obviously, it can be seen from the definition that 𝑘∈Ω₁ is equal to the occurrence of event 𝑑(𝑘)∈[0,𝑑₁) and 𝑘∈Ω₂ means that the event 𝑑(𝑘)∈[𝑑₁,𝑑₂] occurs. Therefore, a stochastic variable 𝛽(𝑘) can be defined as 𝛽(𝑘)=1,𝑘∈Ω₁0,𝑘∈Ω₂. (4)

Assumption 2. 𝛽(𝑘) is a Bernoulli distributed sequence with Prob𝛽(𝑘)=1=𝔼𝛽(𝑘)=𝛽₀,Prob𝛽(𝑘)=0=1−𝔼𝛽(𝑘)=1−𝛽₀, (5) where 0≤𝛽₀≤1 is a constant.

Remark 3. From Assumption 2, it is easy to see that 𝔼{𝛽(𝑘)−𝛽₀}=0 and 𝔼{(𝛽(𝑘)−𝛽₀)²}=𝛽₀(1−𝛽₀). As Prob{𝑑(𝑘)∈[0,𝑑₁)}=Prob{𝛽(𝑘)=1}=𝛽₀ and Prob{𝑑(𝑘)∈[𝑑₁,𝑑₂]}=Prob{𝛽(𝑘)=0}=1−𝛽₀, 𝛽₀ and 1−𝛽₀ also denote the probability of 𝑑(𝑘) taking values in [0,𝑑₁) and [𝑑₁,𝑑₂], respectively.
According to Assumptions 1 and 2, the system model described by (2) can be rewritten as 𝑥_𝑐(𝑘+1)=𝐴_𝑐𝑥_𝑐(𝑘)+𝛽(𝑘)𝐴_𝑑𝑠𝑘−𝑑₁(𝑘)+1−𝛽(𝑘)𝐴_𝑑𝑠𝑘−𝑑₂(𝑘)+𝐵_𝑐𝑣(𝑘),𝑦(𝑘)=𝐶_𝑐𝑥_𝑐(𝑘)+𝛽(𝑘)𝐶_𝑑𝑠𝑘−𝑑₁(𝑘)+1−𝛽(𝑘)𝐶_𝑑𝑠𝑘−𝑑₂(𝑘)+𝐷_𝑐𝑣(𝑘). (6) At the receiving end, we are interested in designing a linear filter with state-realization as follows: Σ_𝑓∶𝑥_𝑓(𝑘+1)=𝐴_𝑓𝑥_𝑓(𝑘)+𝐵_𝑓𝑦(𝑘)𝑠̂(𝑘)=𝐶_𝑓𝑥_𝑓(𝑘)+𝐷_𝑓𝑦(𝑘), (7) where 𝑥_𝑓(𝑘)∈ℜ^𝑛𝑓 is the filter state vector, 𝑠(̂𝑘), is the estimated signal of source signal 𝑠(𝑘) and 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, 𝐷_𝑓 have the following form: 𝐴_𝑓=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0_{𝑛𝑓×𝑛𝑓},𝐵_𝑓=00⋮01_𝑛𝑓×1,𝐶_𝑓=𝑓𝑛_𝑓𝑓𝑛_𝑓−1…𝑓(1)𝐷_𝑓=𝑓(0). (8) The transfer function of the filter is given by Φ_𝑓(𝑧)=𝐶_𝑓𝑧𝐼−𝐴_𝑓⁻¹𝐵_𝑓+𝐷_𝑓=𝑓(0)+𝑓(1)𝑧⁻¹+𝑓(2)𝑧⁻²+⋯+𝑓𝑛_𝑓𝑧^−𝑛𝑓, (9) where 𝑓(0), 𝑓(1),…, and 𝑓(𝑛_𝑓) are parameters to be determined. Define the filtering error as 𝑒(𝑘)=𝑠(𝑘)−𝑠(̂𝑘). Then, via augmenting the models Σ_𝑙 and Σ_𝑐, the filtering error system is given as follows: Σ_𝑒∶𝑥_𝑒(𝑘+1)=𝐴_𝑒𝑥_𝑒(𝑘)+𝛽(𝑘)𝐴_𝑒𝑑𝑥_𝑒𝑘−𝑑₁(𝑘)+1−𝛽(𝑘)𝐴_𝑒𝑑𝑥_𝑒𝑘−𝑑₂(𝑘)+𝐵_𝑒𝑤_𝑒(𝑘)𝑒(𝑘)=𝐶_𝑒𝑥_𝑒+𝛽(𝑘)𝐶_𝑒𝑑𝑥_𝑒𝑘−𝑑₁(𝑘)+1−𝛽(𝑘)𝐶_𝑒𝑑𝑥_𝑒𝑘−𝑑₂(𝑘)+𝐷_𝑒𝑤_𝑒(𝑘), (10) where 𝑥^𝑇_𝑒(𝑘)=𝑥^𝑇_𝑙(𝑘)𝑥^𝑇_𝑐(𝑘)𝑥^𝑇_𝑓(𝑘)^𝑇,𝑤_𝑒(𝑘)=𝑤^𝑇(𝑘)𝑣^𝑇(𝑘)^𝑇,𝐴_𝑒=𝐴_𝑙000𝐴_𝑐00𝐵_𝑓𝐶_𝑐𝐴_𝑓,𝐵_𝑒=𝐵_𝑙00𝐵_𝑐0𝐵_𝑓𝐷_𝑐,𝐶_𝑒=𝐶_𝑙−𝐷_𝑓𝐶_𝑐−𝐶_𝑓,𝐷_𝑒=0−𝐷_𝑓𝐷_𝑐,𝐴_𝑒𝑑=𝐴000_𝑑𝐶_𝑙𝐵00_𝑓𝐶_𝑑𝐶_𝑙00,𝐶_𝑒𝑑=−𝐷_𝑓𝐶_𝑑𝐶_𝑙00. (11) Before giving the main results, we need following definitions at first.

Definition 4. For a given function 𝑉(𝑥(𝑘)), its stochastic difference operator is defined as Δ𝑉(𝑥(𝑘))=𝔼{𝑉(𝑥(𝑘+1))∣𝑥(𝑘)}−𝑉(𝑥(𝑘)). (12)

Definition 5 (see [32]). The filtering error system in (10) is said to be stochastically stable if for any initial condition 𝑥_𝑒(0) and zero exogenous noise 𝑤_𝑒(𝑘), there exists a positive definite 𝑊 independent of 𝑥_𝑒(0), such that the following condition is satisfied: 𝔼^∞_𝑘=0𝑥_𝑒(𝑘)²∣𝑥_𝑒(0)<𝑥^𝑇_𝑒(0)𝑊𝑥_𝑒(0). (13)

Definition 6. System (10) is said to be stochastically stable with an ℋ_∞ norm bound 𝛾, if the following conditions hold.(1)The filtering error system with 𝑤_𝑒(𝑘)=0 is stochastically stable.(2)For all nonzero 𝑤_𝑒(𝑘)∈𝑙₂[0,∞) and under zero initial conditions, the following inequality holds: ‖𝑒(𝑘)‖₂≤𝛾𝑤_𝑒(𝑘)₂. (14)
Now, with the definitions above, we present the objective of this paper.
Given the filtering system shown in Figure 1, we are interested in designing a filter in the form of (7)-(8) such that (a)the filtering error system (10) is asymptotically stable in the stochastic sense;(b)the filtering error system (10) possesses a minimized ℋ_∞ performance level 𝛾;(c)a time-domain envelope constraint is imposed on the output signal 𝑠(̂𝑘) as follows: 𝑙(𝑘)≤𝑠̂(𝑘)≤𝑢(𝑘), (15)where 𝑙(𝑘) and 𝑢(𝑘) are the lower and upper bounds of the time-domain mask, respectively.

3. Main Results

In this section, based on the Lyapunov-Krasovskii stability theorem, a delay-probability-distribution-dependent approach is proposed to solve the ℋ_∞ FIR filter design problem subject to envelope constraints described in (15). First, a stability criterion for the filtering error system described in (10) is proposed. Then the envelope constraints are taken into consideration. An ℋ_∞ optimal ECFIR filter design approach is given at last.

Theorem 7. Given the system in Figure 1, for some given constants 0≤𝑑₁≤𝑑₂, 𝛽₀, and 𝛾, the filtering error system (10) is stochastically stable with ℋ_∞ performance 𝛾 if there exist matrices 𝑃>0, 𝑄₁>0, 𝑄₂>0, 𝑅₁>0, 𝑅₂>0 of appropriate dimensions such that the following optimization problem has solutions, min_{𝑃>0,𝑄1>0,𝑄2>0,𝑅1>0,𝑅2>0,𝑓}𝛾, (16) subject to the following LMI constraint: ΞΞ=¹¹Ξ₁₂Ξ₁₃∗Ξ₂₂Ξ₂₃∗∗Ξ₃₃<0, (17) where Ξ₁₁=−𝑃𝑃𝐴_𝑒𝛽₀𝑃𝐴_𝑒𝑑1∗𝑄−𝑃−𝑑₁𝑅₁1𝑑₁𝑅₁∗∗−𝑄₁−1𝑑₁𝑅₁−1𝑑₂−𝑑₁𝑅₂,Ξ₁₂=1−𝛽₀𝑃𝐵_𝑒𝑃𝐵_𝑒000𝑑₁𝐴^𝑇_𝑒−𝐼𝑅₁1𝑑₂−𝑑₁𝑅₂0𝑑₁𝛽₀𝐴^𝑇_𝑒𝑑𝑅₁,Ξ₁₃=00𝑑₂−𝑑₁𝐴^𝑇_𝑒−𝐼𝑅₂𝐶^𝑇_𝑒𝛽₀𝑑₂−𝑑₁𝐴^𝑇_𝑒𝑑𝑅₂𝛽₀𝐶^𝑇_𝑒𝑑,Ξ₂₂=−𝑄₂−1𝑑₂−𝑑₁𝑅₂0𝑑₁1−𝛽₀𝐴^𝑇_𝑒𝑑𝑅₁∗−𝛾²𝐼𝑑₁𝐵^𝑇_𝑒𝑅₁∗∗−𝑅₁,Ξ₂₃=𝑑₂−𝑑₁1−𝛽₀𝐴^𝑇_𝑒𝑑𝑅₂1−𝛽₀𝐶^𝑇_𝑒𝑑𝑑₂−𝑑₁𝐵^𝑇_𝑒𝑅₂𝐷^𝑇_𝑒,Ξ00₃₃=−𝑅₂0,∗−𝐼𝑄=1+𝑑₁𝑄₁+𝑑₂−𝑑₁+1𝑄₂, (18) and 𝐴_𝑒, 𝐴_𝑒𝑑, 𝐵_𝑒, 𝐶_𝑒, 𝐶_𝑒𝑑, and 𝐷_𝑒 are defined in (11).

Proof. First, define a Lyapunov-Krasovskii functional as follows: 𝑉(𝑘)≜𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘), (19) where 𝑉₁(𝑘)≜𝑥^𝑇_𝑒(𝑘)𝑃𝑥_𝑒(𝑘),𝑉₂(𝑘)≜^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝑥^𝑇_𝑒(𝑖)𝑄₁𝑥_𝑒(𝑖)+^𝑘−1_{𝑖=𝑘−𝑑2(𝑘)}𝑥^𝑇_𝑒(𝑖)𝑄₂𝑥_𝑒(𝑖),𝑉₃(𝑘)≜⁻¹_{𝑖=−𝑑1+2}^𝑘−1_{𝑗=𝑘+𝑖−1}𝑥^𝑇_𝑒𝑗𝑄₁𝑥_𝑒𝑗+^−𝑑1+1_{𝑖=−𝑑2+2}^𝑘−1_{𝑗=𝑘+𝑖−1}𝑥^𝑇_𝑒𝑗𝑄₂𝑥_𝑒𝑗,𝑉₄(𝑘)≜^𝑘−1_{𝑖=𝑘−𝑑1}^𝑘−1_𝑗=𝑖𝛿^𝑇𝑗𝑅₁𝛿𝑗+^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}^𝑘−1_𝑗=𝑖𝛿^𝑇𝑗𝑅₂𝛿𝑗,𝛿𝑗≜𝑥_𝑒𝑗+1−𝑥_𝑒𝑗, (20) and 𝑃=𝑃^𝑇>0, 𝑄₁=𝑄^𝑇₁>0, 𝑄₂=𝑄^𝑇₂>0, 𝑅₁=𝑅^𝑇₁>0, and 𝑅₂=𝑅^𝑇₂>0 are Lyapunov matrices to be determined.
Then using the stochastic difference operator defined in (12), we obtain Δ𝑉₁(𝑘)=𝑥^𝑇_𝑒(𝑘)𝐴^𝑇_𝑒+𝛽₀𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)𝐴^𝑇_𝑒𝑑+1−𝛽₀𝑥^𝑇_𝑒𝑘−𝑑₂(𝑘)𝐴^𝑇_𝑒𝑑+𝑤^𝑇_𝑒(𝑘)𝐵^𝑇_𝑒×𝑃𝐴_𝑒𝑥_𝑒(𝑘)+𝛽₀𝐴_𝑒𝑑𝑥_𝑒𝑘−𝑑₁(𝑘)+1−𝛽₀𝐴_𝑒𝑑𝑥_𝑒𝑘−𝑑₂(𝑘)+𝐵_𝑒𝑤_𝑒(𝑘)−𝑥^𝑇_𝑒(𝑘)𝑃𝑥_𝑒(𝑘),Δ𝑉₂(𝑘)=𝑥^𝑇_𝑒(𝑘)𝑄₁+𝑄₂𝑥_𝑒(𝑘)−𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)×𝑄₁𝑥_𝑒𝑘−𝑑₁(𝑘)−𝑥^𝑇_e𝑘−𝑑₂(𝑘)𝑄₂𝑥_𝑒𝑘−𝑑₂(𝑘)+^𝑘−1_{𝑖=𝑘+1−𝑑1()𝑘+1}𝑥^𝑇_𝑒(𝑖)𝑄₁𝑥_𝑒(𝑖)−^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)+1}𝑥^𝑇_𝑒(𝑖)𝑄₁𝑥_𝑒(𝑖)+^𝑘−1_{𝑖=𝑘+1−𝑑2()𝑘+1}𝑥^𝑇_𝑒(𝑖)𝑄₂𝑥_𝑒(𝑖)−^𝑘−1_{𝑖=𝑘−𝑑2(𝑘)+1}𝑥^𝑇_𝑒(𝑖)𝑄₂𝑥_𝑒(𝑖)≤𝑥^𝑇_𝑒(𝑘)𝑄₁+𝑄₂𝑥_𝑒(𝑘)−𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)×𝑄₁𝑥_𝑒𝑘−𝑑₁(𝑘)−𝑥^𝑇_𝑒𝑘−𝑑₂(𝑘)𝑄₂𝑥_𝑒𝑘−𝑑₂(𝑘)+^𝑘_{𝑖=𝑘−𝑑1+1}𝑥^𝑇_𝑒(𝑖)𝑄₁𝑥_𝑒(𝑖)+^𝑘−𝑑1_{𝑘−𝑑2+1}𝑥^𝑇_𝑒(𝑖)𝑄₂𝑥_𝑒(𝑖),Δ𝑉₃(𝑘)=𝑑₁𝑥^𝑇_𝑒(𝑘)𝑄₁𝑥_𝑒(𝑘)+𝑑₂−𝑑₁𝑥^𝑇_𝑒(𝑘)𝑄₂𝑥_𝑒(𝑘)−^𝑘_{𝑖=𝑘−𝑑1+1}𝑥^𝑇_𝑒(𝑖)𝑄₁𝑥_𝑒(𝑖)−^𝑘−𝑑1_{𝑖=𝑘−𝑑2+1}𝑥^𝑇_𝑒(𝑖)𝑄₂𝑥_𝑒(𝑖),Δ𝑉₄(𝑘)=𝑑₁𝛿^𝑇(𝑘)𝑅₁𝛿(𝑘)+𝑑₂−𝑑₁𝛿^𝑇(𝑘)𝑅₂𝛿(𝑘)−^𝑘−1_{𝑖=𝑘−𝑑1}𝛿^𝑇(𝑖)𝑅₁𝛿(𝑖)−^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛿^𝑇(𝑖)𝑅₂𝛿(𝑖)≤𝑑₁𝛿^𝑇(𝑘)𝑅₁𝛿(𝑘)+𝑑₂−𝑑₁𝛿^𝑇(𝑘)𝑅₂𝛿(𝑘)−^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿^𝑇(𝑖)𝑅₁𝛿(𝑖)−^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿^𝑇(𝑖)𝑅₂𝛿(𝑖). (21) Using the Jensen inequality [33], the following expressions are obtained: −^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿^𝑇(𝑖)𝑅₁𝛿(𝑖)1≤−𝑑₁(𝑘)^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿^𝑇(𝑖)𝑅₁^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿(𝑖)1≤−𝑑₁^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿^𝑇(𝑖)𝑅₁^𝑘−1_{𝑖=𝑘−𝑑1(𝑘)}𝛿(𝑖),−^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿^𝑇(𝑖)𝑅₂𝛿(𝑖)1≤−𝑑₂(𝑘)−𝑑₁^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿^𝑇(𝑖)𝑅₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿(𝑖)1≤−𝑑₂−𝑑₁^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿^𝑇(𝑖)𝑅₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2(𝑘)}𝛿(𝑖). (22) Thus, we have Δ𝑉₄(𝑘)≤𝛿^𝑇(𝑘)𝑑₁𝑅₁+𝑑₂−𝑑₁𝑅₂𝛿(𝑘)−1𝑑₁𝑥^𝑇_𝑒(𝑘)−𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)×𝑅₁𝑥_𝑒(𝑘)−𝑥_𝑒𝑘−𝑑₁(𝑘)−1𝑑₂−𝑑₁𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)−𝑥^𝑇_𝑒𝑘−𝑑₂(𝑘)×𝑅₁𝑥_𝑒𝑘−𝑑₁(𝑘)−𝑥_𝑒𝑘−𝑑₂(𝑘). (23) Thus, we obtain Δ𝑉(𝑘)=Δ𝑉₁(𝑘)+Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘)+Δ𝑉₄(𝑘)≤𝜂^𝑇(𝑘)Υ𝜂(𝑘), (24) where 𝜂^𝑇(𝑘)=𝑥^𝑇_𝑒(𝑘)𝑥^𝑇_𝑒𝑘−𝑑₁(𝑘)𝑥^𝑇_𝑒𝑘−𝑑₂(𝑘)𝑤^𝑇_𝑒(𝑘),ΥΥ=¹¹Υ₁₂Υ₁₃Υ₁₄∗Υ₂₂Υ₂₃Υ₂₄∗∗Υ₃₃Υ₃₄∗∗∗Υ₄₄,Υ₁₁=𝐴^𝑇_𝑒𝑃𝐴_𝑒+𝐴^𝑇_𝑒−𝐼𝑅𝐴_𝑒1−𝐼+𝑄−𝑃−𝑑₁𝑅₁,Υ₁₂=𝛽₀𝐴^𝑇_𝑒𝑃𝐴_𝑒𝑑+𝛽₀𝐴^𝑇_𝑒−𝐼𝑅𝐴_𝑒𝑑+1𝑑₁𝑅₁,Υ₁₃=1−𝛽₀𝐴^𝑇_𝑒𝑃𝐴_𝑒𝑑+1−𝛽₀𝐴^𝑇_𝑒−𝐼𝑅𝐴_𝑒𝑑,Υ₁₄=𝐴^𝑇_𝑒𝑃𝐵_𝑒+𝐴^𝑇_𝑒−𝐼𝑅𝐵^𝑇_𝑒,Υ₂₂=𝛽²₀𝐴^𝑇_𝑒𝑑(𝑃+𝑅)𝐴_𝑒𝑑−𝑄₁−1𝑑₁𝑅₁−1𝑑₂−𝑑₁𝑅₂,Υ₂₃=𝛽₀1−𝛽₀𝐴^𝑇_𝑒𝑑(𝑃+𝑅)𝐴_𝑒𝑑+1𝑑₂−𝑑₁𝑅₂,Υ₂₄=𝛽₀𝐴^𝑇_𝑒𝑑(𝑃+𝑅)𝐵_𝑒,Υ₃₃=1−𝛽₀²𝐴^𝑇_𝑒𝑑(𝑃+𝑅)𝐴_𝑒𝑑−𝑄₂−1𝑑₂−𝑑₁𝑅₂,Υ₃₄=1−𝛽₀𝐴^𝑇_𝑒𝑑(𝑃+𝑅)𝐵_𝑒,Υ₄₄=𝐵^𝑇_𝑒(𝑃+𝑅)𝐵_𝑒,𝑅=𝑑₁𝑅₁+𝑑₂−𝑑₁𝑅₂. (25) By Schur complement, it can be concluded from (17) that Υ<0. By similar lines as in [32], the stochastic stability can be guaranteed if condition (17) holds.
Then, define the performance index as follows: 𝐽=^∞_𝑘=0𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝑤^𝑇_𝑒(𝑘)𝑤_𝑒(𝑘). (26) Considering the fact that 𝑉(𝑘)≥0, under the zero initial condition, we have 𝐽≤^∞_𝑘=0𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝑤^𝑇_𝑒(𝑘)𝑤_𝑒(𝑘(∞)(0)=)+𝑉−𝑉_∞𝑘=0𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝑤^𝑇_𝑒(𝑘)𝑤_𝑒(𝑘)+Δ𝑉(𝑘). (27) Thus, 𝐽<0 is equal to 𝜂^𝑇(𝑘)(Θ+Υ)𝜂(𝑘)<0, (28) where𝐶Θ=^𝑇𝑒𝐶_𝑒𝛽₀𝐶^𝑇_𝑒𝐶_𝑒𝑑1−𝛽₀𝐶^𝑇_𝑒𝐶_𝑒𝑑𝐶^𝑇_𝑒𝐷_𝑒∗𝛽²₀𝐶^𝑇_𝑒𝑑𝐶_𝑒𝑑𝛽₀1−𝛽₀𝐶^𝑇_𝑒𝑑𝐶_𝑒𝑑𝛽₀𝐶^𝑇_𝑒𝑑𝐷_𝑒∗∗1−𝛽₀²𝐶^𝑇_𝑒𝑑𝐶_𝑒𝑑1−𝛽₀𝐶^𝑇_𝑒𝑑𝐷_𝑒∗∗∗𝐷^𝑇_𝑒𝐷_𝑒−𝛾²𝐼. (29)Through applying Schur complement, it is shown that (Θ+Υ)<0 can be guaranteed by condition (17). That is to say, once (17) is satisfied, the ℋ_∞ performance can be guaranteed to be less than 𝛾. Thus, the proof is completed.
At this point, the second desired property of the system will be considered, which is the envelope constraints demand. First, some notations are introduced [34]: 𝑦(0)𝑦(1)⋮𝑦(𝑚)𝑙(0)𝑙(1)⋮𝑙(𝑛),𝑢(0)𝑢(1)⋮𝑢(𝑛)𝑓(0)𝑓(1)⋮𝑓𝑛𝑦=,𝑙=𝑢=,𝑓=_𝑓,𝑦(0)𝑦(1)𝑦(0)(1)𝑦(𝑚)(0)(𝑚)(1)(𝑚)𝑌=,0⋯0⋯0⋮𝑦⋯⋮⋮𝑦0𝑦⋮𝑦⋮⋮⋮00⋯𝑦 (30) where 𝑌 is an 𝑛×(𝑛_𝑓+1) matrix, 𝑛=𝑚+𝑛_𝑓+1, 𝑦(0)𝑦(1)⋯𝑦(𝑚)00⋯ (31) is a given signal, and 𝑙(0)𝑙(1)(𝑚),𝑢(0)𝑢(1)(𝑚)⋯𝑙⋯𝑢 (32) Are, respectively, the upper and lower bounds. Therefore, the constraint of (15) is equal to diag(𝑙)≤diag𝑌𝑓≤diag(𝑢), (33) where diag(•) denotes a conversion from a vertical vector to a diagonal matrix.
Based on Theorem 7 and (33), we can establish another theorem to determine the filter that satisfies the envelope constraint meanwhile possessing optimal ℋ_∞ performance.

Theorem 8. An ℋ_∞ optimal filter of the form (7)-(8) satisfying envelope constraint in (15) can be obtained by solving the following LMI optimization problem: min_{𝑃>0,𝑄1>0,𝑄2>0,𝑅1>0,𝑅2>0,𝑓}𝛾, (34) subject to ΞΞ=¹¹Ξ₁₂Ξ₁₃∗Ξ₂₂Ξ₂₃∗∗Ξ₃₃(𝑙),(𝑢),<0,diag≤diag𝑌𝑓diag𝑌𝑓≤diag (35) where Ξ is defined in (17).

4. An Illustrative Example

In this section, an example is given to support the filter design method proposed in the paper. Consider a filtering system as shown in Figure 1. The parameters for Σ_𝑙 are given by 𝐴_𝑙=,𝐵−2.3060−2.9625−2.2590−1.0922−0.3009−0.0325100000010000001000000100000010_𝑙=100000,𝐶_𝑙=00000.00620.2170. (36)

The parameters for the delay channel Σ_𝑐 are given by 𝐴_𝑐=010−0.1,𝐴_𝑑=00.1,𝐵_𝑐=,𝐶0.10.1_𝑐=01,𝐶_𝑐𝑑=0.2,𝐷_𝑐𝑑=1,₁=2,𝑑₂=3,𝛽₀=0.7. (37) Using Theorem 8, the ℋ_∞ optimal filter is obtained via using the LMI toolbox of MATLAB with 𝑛_𝑓 chosen to be 5. The resultant optimal 𝛾 is 8.5057 and filter gains are given as follows: 𝐶_𝑓=,𝐷0.0437−0.3344−0.40230.2045−0.5089_𝑓=4.3664. (38)

The expected envelope constraints and 𝑠(𝑘) (the output of Σ_𝑙) corresponding to a particular case where input signal 𝑤(𝑘) is chosen to be unit impulse signal are shown in Figure 2. The transmitted signal 𝑦(𝑘) through Σ_𝑐 which is generated with no noise added is also given in the figure. The filter output 𝑠(̂𝑘) and filtering error 𝑒(𝑘) are given in Figures 3 and 4, respectively.

Figure 2 

Source signal 𝑠(𝑘), filter input signal 𝑦(𝑘), and envelope bounds.

Figure 3 

Output of the filter without disturbance.

Figure 4 

Filtering error 𝑒(𝑘) without disturbance.

Furthermore, to illustrate the performance of the designed filter, we add the disturbance signal 𝑣(𝑘) chosen as white noise with mean of zero and variance of 1×10⁻³ into the system. The resultant filter output and filtering error are shown in Figures 5 and 6, respectively. It is shown that the designed filter is effective.

Figure 5 

Output of the filter with disturbance.

Figure 6 

Filtering error 𝑒(𝑘) with disturbance.

5. Conclusions

In this paper, we have solved the filtering design problem of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the envelope constraints are also considered in the process of filtering design. The delay-distribution-dependent criterion is formed for the filtering error system, employing the information about not only the size of delay but also its probability distribution. A set of linear matrix inequalities (LMIs) are formulated to solve the problem. Through solving the LMI optimization problem, the ℋ_∞ performance is minimized and pulse-shape demand imposed by envelope constraints is satisfied. Finally, an illustrative example is presented to demonstrate the effectiveness of the filtering design approach. For future research directions, extending the filter design approach proposed in this paper to networked control systems and distributed systems is an interesting issue. Besides, more general filter design approaches considering delays in different forms with different characteristics also deserve further investigation.

Acknowledgments

The authors are grateful to the associate editor and anonymous reviewers for their constructive comments based on which the presentation of this paper has been greatly improved.